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On $\overline{\partial }_{b}$-harmonic maps from pseudo-Hermitian manifolds to Kähler manifolds

Yuxin Dong, Hui Liu, Biqiang Zhao

TL;DR

The paper addresses extending strong rigidity phenomena for harmonic maps to the CR/pseudo-Hermitian setting by introducing partial energy functionals $E_{\bar\partial_b,\xi}$ and $E_{\partial_b,\xi}$ that measure horizontal and Reeb-direction contributions. It develops foliated Lichnerowicz-type invariants, derives comprehensive commutation relations and a Paneitz-type operator to manage mixed terms, and proves rigidity results under curvature and rank assumptions, yielding foliated CR-holomorphicity for $\bar\partial_b$- and $\partial_b$-harmonic maps. The results generalize Petit's foliated rigidity and Siu-type holomorphicity to general pseudo-Hermitian domains and provide CR-analogue tools (e.g., $Pf$, $B_f$, $E$ tensors) for controlling the geometry of maps into Kähler targets with negative curvature. The work deepens the understanding of pseudoharmonic maps, establishing foliated holomorphicity and pluriharmonicity as natural outcomes under curvature constraints, with potential implications for CR geometry and rigidity phenomena on pseudo-Hermitian manifolds.

Abstract

In this paper, we consider maps from pseudo-Hermitian manifolds to Kähler manifolds and introduce partial energy functionals for these maps. First, we obtain a foliated Lichnerowicz type result on general pseudo-Hermitian manifolds, which generalizes a related result on Sasakian manifolds in \cite{SSZ2013holomorphic}. Next, we investigate critical maps of the partial energy functionals, which are referred to as $\overline{\partial }_{b}$-harmonic maps and $\partial _{b}$-harmonic maps. We give a foliated result for both $\overline{\partial }_{b}$- and $\partial _{b}$-harmonic maps, generalizing a foliated result of Petit \cite{Pet2002harmonic} for harmonic maps. Then we are able to generalize Siu's holomorphicity result for harmonic maps \cite{Siu1980rigid} to the case for $\overline{\partial }_{b}$- and $\partial _{b}$-harmonic maps.

On $\overline{\partial }_{b}$-harmonic maps from pseudo-Hermitian manifolds to Kähler manifolds

TL;DR

The paper addresses extending strong rigidity phenomena for harmonic maps to the CR/pseudo-Hermitian setting by introducing partial energy functionals and that measure horizontal and Reeb-direction contributions. It develops foliated Lichnerowicz-type invariants, derives comprehensive commutation relations and a Paneitz-type operator to manage mixed terms, and proves rigidity results under curvature and rank assumptions, yielding foliated CR-holomorphicity for - and -harmonic maps. The results generalize Petit's foliated rigidity and Siu-type holomorphicity to general pseudo-Hermitian domains and provide CR-analogue tools (e.g., , , tensors) for controlling the geometry of maps into Kähler targets with negative curvature. The work deepens the understanding of pseudoharmonic maps, establishing foliated holomorphicity and pluriharmonicity as natural outcomes under curvature constraints, with potential implications for CR geometry and rigidity phenomena on pseudo-Hermitian manifolds.

Abstract

In this paper, we consider maps from pseudo-Hermitian manifolds to Kähler manifolds and introduce partial energy functionals for these maps. First, we obtain a foliated Lichnerowicz type result on general pseudo-Hermitian manifolds, which generalizes a related result on Sasakian manifolds in \cite{SSZ2013holomorphic}. Next, we investigate critical maps of the partial energy functionals, which are referred to as -harmonic maps and -harmonic maps. We give a foliated result for both - and -harmonic maps, generalizing a foliated result of Petit \cite{Pet2002harmonic} for harmonic maps. Then we are able to generalize Siu's holomorphicity result for harmonic maps \cite{Siu1980rigid} to the case for - and -harmonic maps.

Paper Structure

This paper contains 5 sections, 17 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lichnerowicz type results
  4. Commutation relations
  5. Foliated and $(J,\widetilde{J})$-holomorphicity results

Key Result

Theorem 1.1

Let $(M^{2m+1},H,J,\theta )$ be a compact pseudo-Hermitian manifold, and $(N,\widetilde{J},\widetilde{g})$ be a Kähler manifold. Then $K_{b}(f)$ is a smooth foliated homotopy invariant, that is, $K_{b}(f_{t})$ is constant for any family $\{f_{t}\}$ of foliated maps.

Theorems & Definitions (34)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1: CDRY2019onharm
  • Definition 2.2
  • Definition 2.3: GIP2001cr
  • Remark 2.1
  • Definition 2.4
  • Remark 2.2
  • Lemma 2.5
  • ...and 24 more